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We revisit the derivation of the expressions for the calculation of second-harmonic generation spectra and show the equivalence of different approaches proposed in the literature. A method is suggested to include excitonic effects in the calculation. Numerical tests are performed for the linear and nonlinear optical response of bulk GaAs. Excitonic effects are found to give rise to a redistribution of oscillator strength, thus improving the agreement with experiment.
Leitsmann et al. (Tue,) studied this question.